"I have a fever, but don't have a cough."
In colloquial Mandarin 有 is sometimes used as a perfect marker similar to 了. But I associate that usage specifically with Taiwanese Mandarin (it's a Minnan borrowing, so maybe maybe it's also used like that in the area around Xiamen. Can't confirm that first-hand though). In Standard 普通话 you're correct.
It's acceptable. Add 有 before 发烧 is common in southern China and means the same.
There are still more sentences with 有 + verb structure. Unfortunately the course doesn't explain it at all.
Yeah, I've never heard it with you3 before... Sometimes I feel like some Taiwanese person wrote some of these - or it's just some weird regional accent.
It can also be used as a noun, at least colloquially. There's nothing wrong with the Chinese sentence here.
Just because it's spoken in south China doesn't mean it's correct. People speak ❤❤❤❤ all the time.